def test_kh_uniform_solution(show=False, verbose=False):
"""Test eigenvalue solver using FourierGrid"""
import numpy as np
from psecas import Solver, FourierGrid
from psecas.systems.kh_uniform import KelvinHelmholtzUniform
grid = FourierGrid(N=64, zmin=0, zmax=2)
system = KelvinHelmholtzUniform(grid, beta=1e4, nu=1e-2, kx=3.52615254237)
system.u0 = 1.
solver = Solver(grid, system)
Ns = np.hstack((np.arange(2, 16) * 32, np.arange(2, 12) * 64))
for useOPinv in [True, False]:
omega, v, err = solver.iterate_solver(Ns,
tol=1e-5,
useOPinv=useOPinv,
verbose=verbose)
np.testing.assert_allclose(1.66548246011, omega, atol=1e-5)
if show:
from psecas import plot_solution
plot_solution(system, smooth=True, num=2)
# Check that information converged is set to False when solver does not converge
solver.iterate_solver(np.arange(1, 3) * 16,
tol=1e-16,
useOPinv=useOPinv,
verbose=verbose)
assert solver.system.result['converged'] is False
return err
def test_mti_solution(show=False, verbose=False):
"""Test eigenvalue solver using ChebyshevExtremaGrid"""
import numpy as np
from psecas import Solver, ChebyshevExtremaGrid
from psecas.systems.mti import MagnetoThermalInstability
grid = ChebyshevExtremaGrid(N=64, zmin=0, zmax=1)
system = MagnetoThermalInstability(grid, beta=1e5, Kn0=200, kx=4 * np.pi)
system.beta = 1e5
system.Kn0 = 200
assert system.Kn0 == 200
assert system.beta == 1e5
solver = Solver(grid, system)
Ns = np.hstack(np.arange(1, 10) * 16)
mode = 0
omega, vec, err = solver.iterate_solver(Ns,
mode=mode,
tol=1e-8,
verbose=verbose)
system.get_bx_and_by()
if show:
from psecas import plot_solution
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
plot_solution(system, smooth=True, num=1)
np.testing.assert_allclose(1.7814514515967603, omega, atol=1e-8)
return err
Nz=256,
Lz=2.0,
path="./",
name="khi_hydro_delta",
)
# Print out some information
Lx = 2 * np.pi / system.kx
print('')
print('Eigenvalue is:', omega)
print('Lx should be:', Lx)
# Make a plot
plt.figure(1)
plt.plot(kxmax, omega.real, "+")
plot_solution(system, filename='./khi_hydro_delta.pdf')
# Write files for loading into Athena
s = system
c_dic = {}
for key in s.variables:
c_dic.update({key: s.grid.to_coefficients(s.result[key])})
perturb = []
for key in ['drho', 'dvx', 'dvz', 'dT']:
perturb.append(c_dic[key].real)
perturb.append(c_dic[key].imag)
perturb = np.transpose(perturb)
np.savetxt(
for key in system.variables:
system.result[key] /= val
# Save system pickle object
save_system(system, "./khi_nu.p")
# Print out some information
Lx = 2 * np.pi / system.kx
print('')
print('Eigenvalue is:', omega)
print('Lx should be:', Lx)
# Make a plot
plt.figure(1)
plt.plot(kxmax, omega.real, "+")
plot_solution(system, filename='./khi_nu.pdf')
# Write files for loading into Athena
s = system
c_dic = {}
for key in s.variables:
c_dic.update({key: s.grid.to_coefficients(s.result[key])})
perturb = []
for key in ['drho', 'dvx', 'dvz', 'dT', 'dA']:
perturb.append(c_dic[key].real)
perturb.append(c_dic[key].imag)
perturb = np.transpose(perturb)
np.savetxt(
system.add_equation(
"-r*r*sigma*bphi = - 1j*kz*r*r*vphi - 1j*kz*q*r**(2-q)*Aphi - eta*(Drbphi) - lh*va*1j*kz*(DrAphi)"
)
# The boundary conditions
Aphi_bound = 'r**2*dr(dr(Aphi)) + r*dr(Aphi) - Aphi = 0'
system.add_boundary('vr', 'Dirichlet', 'Dirichlet')
system.add_boundary('vphi', 'Dirichlet', 'Dirichlet')
system.add_boundary('vz', 'Neumann', 'Neumann')
system.add_boundary('Aphi', Aphi_bound, Aphi_bound)
system.add_boundary('bphi', 'Dirichlet', 'Dirichlet')
system.add_substitution(
'DrAphi = r*r*dr(dr(Aphi)) + r*dr(Aphi) - Aphi - kz**2*r*r*Aphi')
system.add_substitution(
'Drbphi = r*r*dr(dr(bphi)) + r*dr(bphi) - bphi - kz**2*r*r*bphi')
solver = Solver(grid, system)
mode = 0
Ns = np.arange(1, 32) * 32 - 1
omega, vec, err = solver.iterate_solver(Ns, mode=mode, verbose=True, tol=1e-4)
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
if isinstance(grid, ChebyshevTLnGrid):
plot_solution(system, limits=[0, 1.5])
plt.xlim(0, 1.5)
else:
plot_solution(system)
def test_mri_solution(show=False, verbose=False):
"""Test eigenvalue solver using ChebyshevExtremaGrid"""
import numpy as np
from psecas import Solver, ChebyshevExtremaGrid, System
class HallMRI(System):
def __init__(self, grid, kz, variables, eigenvalue):
# Set parameters
self.q = 1.5
self.eta = 0.003
self.lh = 1
self.h = 0.25
self.va = 0.002
self.kz = kz
super().__init__(grid, variables, eigenvalue)
# Create a grid
grid = ChebyshevExtremaGrid(N=128, zmin=1, zmax=2, z='r')
variables = ['rho', 'vr', 'vphi', 'vz', 'Aphi', 'bphi']
kz = 2 * np.pi
# Create the system
system = HallMRI(grid, kz, variables=variables, eigenvalue='sigma')
# The linearized equations
system.add_equation("-r*sigma*rho = r*dr(vr) + vr + 1j*kz*r*vz")
system.add_equation(
"-r*r*sigma*vr = - 2*r**(2-q)*vphi + h**2*r*r*dr(rho) + va**2*(DrAphi)"
)
system.add_equation("-sigma*vphi = + (2-q)*r**(-q)*vr - va**2*1j*kz*bphi")
system.add_equation("-sigma*vz = h**2*1j*kz*rho")
system.add_equation(
"-r*r*sigma*Aphi = + r*r*vr - eta*(DrAphi) + lh*va*1j*kz*r*r*bphi")
system.add_equation(
"-r*r*sigma*bphi = - 1j*kz*r*r*vphi - 1j*kz*q*r**(2-q)*Aphi - eta*(Drbphi) - lh*va*1j*kz*(DrAphi)"
)
# The boundary conditions
Aphi_bound = 'r**2*dr(dr(Aphi)) + r*dr(Aphi) - Aphi = 0'
system.add_boundary('vr', 'Dirichlet', 'Dirichlet')
system.add_boundary('vphi', 'Dirichlet', 'Dirichlet')
system.add_boundary('vz', 'Neumann', 'Neumann')
system.add_boundary('Aphi', Aphi_bound, Aphi_bound)
system.add_boundary('bphi', 'Dirichlet', 'Dirichlet')
# Short hands for long expressions for derivatives
system.add_substitution(
'DrAphi = r*r*dr(dr(Aphi)) + r*dr(Aphi) - Aphi - kz**2*r*r*Aphi')
system.add_substitution(
'Drbphi = r*r*dr(dr(bphi)) + r*dr(bphi) - bphi - kz**2*r*r*bphi')
solver = Solver(grid, system)
mode = 0
Ns = np.hstack(np.arange(1, 10) * 32)
omega, vec, err = solver.iterate_solver(Ns,
mode=mode,
tol=1e-8,
verbose=verbose)
if show:
from psecas import plot_solution
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
plot_solution(system, smooth=True, num=1)
np.testing.assert_allclose(0.09892641, omega, atol=1e-8)
return err
import numpy as np
from psecas import Solver, LegendreExtremaGrid
import matplotlib.pyplot as plt
from psecas import plot_solution
grid = LegendreExtremaGrid(N=200, zmin=-0.5, zmax=0.5)
system = KelvinHelmholtzMiura(grid, kx=2*np.pi, B=0.2, z1=0, a=1/25.0)
solver = Solver(grid, system)
Ns = np.arange(1, 8) * 64
omega, v, err = solver.iterate_solver(Ns, verbose=True, tol=1e-8)
print('\nB={:1.2f} has gammma*2a = {:1.6f}'.format(system.B,
omega.real*system.a*2))
plot_solution(system, num=1, filename='Ryu_and_Frank1996.pdf')
# Make figure 4 in Miura & Pritchett (1982)
# This calculation takes some time.
if True:
# The iterative solver is more precise but slow at high kx
# Set fast=True to instead run with a fixed grid size and get a figure
# much faster. The result looks okay but is imprecise at high k!
fast = False
plt.figure(2)
plt.clf()
for B in [1e-8, 0.2, 0.3, 0.4]:
system = KelvinHelmholtzMiura(grid, 2*np.pi, B=B, z1=0, a=1/20.0)
beta = 1e5
Kn = 1 / 1500.
kx = 250
system = HeatFluxDrivenBuoyancyInstability(grid, beta, Kn, kx)
solver = Solver(grid, system)
mode = 2
Ns = np.hstack((np.arange(2, 5) * 16, np.arange(3, 12) * 32))
omega, vec, err = solver.iterate_solver(Ns, mode=mode, verbose=True, tol=1e-5)
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=mode)
plot_solution(system, smooth=True)
if True:
# Plot 2D maps of the perturbations
import matplotlib.pyplot as plt
from psecas import get_2Dmap
plt.rc("image", origin="lower", cmap="RdBu")
plt.figure(2)
plt.clf()
fig, axes = plt.subplots(num=2, sharex=True, sharey=True, ncols=4)
xmin = 0
xmax = grid.zmax/10
Nx = 512
Nz = 1024
drhodz_sym = sym.diff(rho_sym, z)
domgdz_sym = sym.diff(Omg_sym, z)
zg = self.grid.zg
self.rho = np.ones_like(zg) * lambdify(z, rho_sym)(zg)
self.Omg = np.ones_like(zg) * lambdify(z, Omg_sym)(zg)
self.shr = np.ones_like(zg) * lambdify(z, shr_sym)(zg)
self.drhodz = np.ones_like(zg) * lambdify(z, drhodz_sym)(zg)
self.domgdz = np.ones_like(zg) * lambdify(z, domgdz_sym)(zg)
# Create a grid
grid = ChebyshevRationalGrid(N=200, C=0.2)
# grid = ChebyshevExtremaGrid(N=199, zmin=-5, zmax=5)
# Create the system
system = VerticalShearInstability(grid,
variables=['rh', 'wx', 'wy', 'wz'],
eigenvalue='sigma')
# The linearized equations
system.add_equation("-sigma*rh = - 1j*kx*wx - 1/h*dz(wz) - 1/h*drhodz/rho*wz")
system.add_equation("-sigma*wx = + 2*Omg*wy - 1j*kx*(h/O0)**2*rh")
system.add_equation("-sigma*wy = - (2*Omg + shr)*wx - 1/h*domgdz*wz")
system.add_equation("-sigma*wz = - h/O0**2*dz(rh)", boundary=True)
solver = Solver(grid, system, do_gen_evp=True)
omega, vec = solver.solve(mode=0, verbose=True)
plot_solution(system)
plt.rc("image", origin="lower", cmap="RdBu", interpolation="None")
extent = [xmin, xmax, zmin, zmax]
phi = np.arctan(vec[2].imag / vec[2].real)
solver.keep_result(omega, vec * np.exp(-1j * phi), mode=0)
# Normalize eigenmodes
y = np.vstack([
system.result["dvx"].real,
system.result["dvx"].imag,
system.result["dvz"].real,
system.result["dvz"].imag,
])
limits = [-2, 2]
plot_solution(system, num=1, limits=limits, smooth=True)
plt.xlim(limits[0], limits[1])
maps = {
key: get_2Dmap(system, key, xmin, xmax, Nx, Nz, zmin=zmin, zmax=zmax)
for key in system.variables
}
plt.figure(2)
plt.clf()
fig, axes = plt.subplots(num=2)
axes.imshow(maps["dT"] + maps["drho"], extent=extent)
for key in system.variables:
system.result[key] /= val
# Save system pickle object
save_system(system, "./khi_mhd_beta5.p")
# Print out some information
Lx = 2 * np.pi / system.kx
print('')
print('Eigenvalue is:', omega)
print('Lx should be:', Lx)
# Make a plot
plt.figure(1)
plt.plot(kxmax, omega.real, "+")
plot_solution(system, filename='./khi_mhd_beta5.pdf')
# Write files for loading into Athena
s = system
c_dic = {}
for key in s.variables:
c_dic.update({key: s.grid.to_coefficients(s.result[key])})
perturb = []
for key in ['drho', 'dvx', 'dvz', 'dT', 'dA']:
perturb.append(c_dic[key].real)
perturb.append(c_dic[key].imag)
perturb = np.transpose(perturb)
np.savetxt(